"Spherate" is a term i invented, to describe what happens when you replace points with balls, and edges with pipes. It includes figures with odd sails representing faces as well. In essence, one replaces thin things with things with a bit of substance for modelling and clarity. The Zome-tool thing makes spherated edge-frames of models, consists of balls which you stick rods into.

Since a circle-edge is a line, you can 'spherate' this too. In the extreme case, you get a thin hoop, but making the spherate larger, the thing becomes more like a car tyre or doughnut. But a torus, be it hoop or doughnut, is itself a solid of interest.

In higher dimensions, even 2-dimensional surfaces are not bounding, so i call these 'hedra' rather than words that suggest division, like 'plane'. You can take a thing like a circle, or a sphere-surface, and spherate it. This is just as in 3d, running a little ball around the figure, and anything the sphere touches is inside it. So a circle perpendicular to a plane in 4d (ie a 3d space), strikes it at two points. If we spherate these two points, we replace these with two little spheres. Rotating this around the whole circle, we get a 'sphere * line' prism, that has been hooked up at the ends, like a hose.

Spheration is a kind of 'modeller's finish', like a paint, applied to thin abstract objects. Other kinds of 'paint' include surtope paint, which replaces smooth surfaces with faces, edges and vertices of a polyhedron. For example, a cube is a sphere painted with surtope paint. More coats of this paint gives more polytope, eg it could come out with hundreds of elements.

FIVE PRODUCTSThe things like rss etc, belong to the domain of the five products. Spheration is not one of them.

The five products produce the five infinite families of regular solids. Three of these five products have expressions in terms of a radiant function, and four are in expressions of adding extra elements to the solid before doing the product.

- pyramid product, produces simplexes, by draught of content, adds a dimension to the product
- tegum product (sum), produces the cross polytope, by draught of surface
- prism product (max), produces the measure polytope, by repetition of content
- crind product (rss), produces spheres
- comb product, produces cubic-lattices, by repetition of surface, reduces a dimension from the product

The radiant products work like this. A solid is regarded as a radial function from some centre, where the centre is taken as zero, and the surface is taken as one. This sets a length measure in every direction. When the radiant product is applied over figures X, Y, Z, ..., these figures are placed orthogonally to each other, surface of the product is read as 1 = f(x,y,z, ... ), where f() is the sum, max, or rss functions as the product is the tegum, cube, or sphere.

The standard line goes from -1 to 1 in 1D. The centre of the line is 0, so the radiant function of the line is then abs(x).

If one takes the product of three standard lines in 3d, one gets the standard tegum, cube and sphere respectively. At a point (x,y,z), the radiant coordinate is what (x), (y), and (z) would become, ie abs(x), abs(y), abs(z). The surface of an octahedron is then defined by the surface sum(abs(x), abs(y),abs(z)) = 1. This is the plane x+y+z=1, x-y+z=1, etc. The surface of a cube is defined by max(abs(x), abs(y), abs(z)) = 1, which means that it's bound by planes x=1, x=-1, y=1, y=-1, z=1, z=-1, and the vertices are +/- 1, +/-1, +/- 1. The sphere is defined by rss(abs(x), abs(y), abs(z)) = 1, ie x²+y²+z² = 1².

All products can be freely applied, so one can for example, take the crind-product of x,y and then the prism product of XY, Z. The surface is then described by the general equation 1 = max( rss(abs(x), abs(y)), abs(z)).

These products form a class of 'brick' polytopes, when applied over the standard line. The products are greatly simplified to <tegum>, [prism], (crind), so the standard octahedron is (x,y,z), the standard cube is [x,y,z], and the standard sphere is (x,y,z). A cylinder can be written as [x,(y,z)]. The intersection of two cylinders [(x,z),y] and [(y,z),x], gives a standard crind ([x,y],z).

TorotopesTorotopes are varieties of round things with holes in them. It's a rather messy process, but involves both the comb product and/or spheration.

The comb product is a repetition of surface. A polygon or circle has a line as a surface. If you multiply a line by a line, you get a rectangle. A polytope bounded by a rectangle is a 3d thing. Let's suppose that you take a sheet of paper. This represents the product of a circle-surface by a circle-edge. You can now hook the left and right hand side of the paper together, to get some sort of hollow cylinder. Now hook the top and bottom, to get a torus.

In four dimensions, the paper has three axies. So after the first two steps, you still have a 'torus-prism'. You can hook the top and bottom together to get a new kind of torus.

The comb product is a product of surfaces, so you are not restricted to surfaces of polygons. You can use polyhedra, etc, as long as the surface dimensions add up to the dimension of the surface of a 4d thing (ie 3). So you can make a dodecahedron - line prism in 4d. We're only interested in the product of surfaces, the pentagonal prisms that make the 'height', not the dodecahedral end-pieces. So we might put a hoop, a thin circle through the dodecahedron faces.

You can then 'bend' the line into a circle, so you get a dodecahedral-circle comb. This would be like stretching it so it goes right around the hoop. This is topological different to the torotopes we met before.

Instead of bending it, you could 'roll it down' as you might take a sock off. The top still meets the bottom, but because it is rolling down outside the prism, the hoop is not covered, and so when the ends hook up, the body of the solid does not include the hoop. It's topologically equal to the previous shape's surface, but the interior is a different shape.

The hose and sock process are identical, but in different directions. So, for example, T( dodeca , line), can be formed by doing a hose process around the dodecahedron, or doing a sock process around the line. The torotope notation is based on this.

Instead of doing a comb product, one can do a series of spherations. For example, the dodecahedron-circle comb, starts off as a point in 4d. Make a sphere around the point, in a 3d space (this spherates the point in a 3-space). Now use a smaller radius, and spherate the surface in 3d + 2d - 1d = 4d. This replaces a point on the surface of a sphere, by a little circle, in the same way that the section of a torus is two small circles.

Going the other way, spherate a point in a 2-space, to get a circle. Now, in 4d, you can spherate the circle, by putting a little sphere at that point.